The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.
So ...

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! ...

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...

I've been wrecking my brain with this problem and I really hope you can help me. You see I have a triangle that is either an isosceles or equilateral or right and I have to find a way to:
1)Convert it ...

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...

What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know ...

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all, I would ...

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...

Don't let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can't think of an easy/smart way of finding it.
Any ideas?
For reference, the Area is:
...

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.
The product of two numbers, $2$ and $3$, we say that it ...

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?
One approach is to find the length of each side from the coordinates ...

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it.
I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...

The area of a circle radius $R$ is $\pi R^2$ which is quite easy to prove with integral calculus.
Consider a ring of radius $\mathrm{d}r$ at a distance $r$ from the centre. This ring has area $2\pi r ...

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?
$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$
A gives us the radius of the circle, B changes the frequency ...

Assuming I have the graph of a function $f(x)$ is there function $f_1(f(x))$ that will give me a rotated version of the graph of that function?
For example if I plot $\sin(x)$ I will get a sine wave ...

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction.
But in 3D I don'...